Radius and profile of random planar maps with faces of arbitrary degrees

We prove some asymptotic results for the radius and the profile of large random rooted planar maps with faces of arbitrary degrees. Using a bijection due to Bouttier, Di Francesco and Guitter between rooted planar maps and certain four-type trees with positive labels, we derive our results from a conditional limit theorem for four-type spatial Galton-Watson trees.


Introduction
This paper is devoted to the proof of limit theorems for random planar maps with no constraint on the degree of faces.This work is a natural sequel to the papers [2,4,8,12,11], which dealt with such limit theorems with an increasing level of generality, starting from the case of planar quadrangulations and moving to invariance principles for the radius and the profile of bipartite, then general, Boltzmann-distributed random planar maps.
Our main goal is to obtain invariance principles for certain functionals of planar maps with no constraint on the face degrees, of the same kind as those obtained in [11].However, while this work focused on rooted and pointed planar maps, with distances measured from the distinguished vertex, we focus on maps that are only rooted and measure distances from the root edge.Similar "rooted" results where obtained in [12] building on the "rootedpointed" results of [8].
The basic tools we rely on -the Bouttier-Di Francesco-Guitter bijection [1] and methods derived from Le Gall's work [4] -are quite close to those of [12].However, there are some notable differences which make the study more intricate.One of the key differences lies in a change in a re-rooting lemma for discrete trees, which is considerably more delicate in the present setting where multiple types are allowed (see Section 3.1).
Our approach in this paper will be to focus essentially on these differences, while the parts which can be derived mutatis mutandis from [4,12] will be omitted.

Preliminaries
2.1.Boltzmann laws on planar maps.A planar map is a proper embedding, without edge crossings, of a connected graph in the 2-dimensional sphere S 2 .Loops and multiple edges are allowed.The set of vertices will always be equipped with the graph distance : if a and a ′ are two vertices, d(a, a ′ ) is the minimal number of edges on a path from a to a ′ .If m is a planar map, we write F m for the set of its faces, and V m for the set of its vertices.
A rooted planar map is a pair (m, e ) where m is a planar map and e is a distinguished oriented edge.The origin of e is called the root vertex.A rooted pointed planar map is a triple (m, τ, e ) where (m, e ) is a rooted planar map and τ is a distinguished vertex.We assume that the vertex map, which is denoted by †, is a rooted pointed planar map.
Two rooted maps (resp.two rooted pointed maps) are identified if there exists an orientation-preserving homeomorphism of the sphere that sends the first map to the second one and preserves the root edge (resp.the distinguished vertex and the root edge).Let us denote by M r (resp.M r,p ) the set of all rooted maps (resp.the set of all rooted pointed maps) up to the preceding identification.In what follows, we will focus on the subset M + r,p of M r,p defined by : M + r,p = {(m, τ, e ) ∈ M r,p : d(τ, e + ) = d(τ, e − ) + 1} ∪ { †}.
Let us recall some definitions that can be found in [11].Let q = (q i , i ≥ 1) be a sequence of nonnegative weights such that q 2κ+1 > 0 for at least one κ ≥ 1.For any planar map m = †, we define W q (m) by W q (m) = where we have written deg(f ) for the degree of the face f , and we set W q ( †) = 1.We require q to be admissible that is W q (m) < ∞.
Set also W q (m).
For k, k ′ ≥ 0 we set . For every weight sequence we define From Proposition 1 in [11], a sequence q is admissible if and only if the system has a solution (z + , z ♦ ) ∈ (0, +∞) 2 for which the matrix M q (z + , z ♦ ) defined by has a spectral radius ̺ ≤ 1. Furthermore this solution is unique and q , where (Z ♦ q ) 2 = Z q − 2Z + q + 1.An admissible weight sequence q is said to be critical if the matrix M q (Z + q , Z ♦ q ) has a spectral radius ̺ = 1.An admissible weight sequence q is said to be regular critical if q is critical and if f • q (Z + q + ε, Z ♦ q + ε) < ∞ for some ε > 0. Let q be a regular critical weight sequence.We define the Boltzmann distribution B + q on the set M + r,p by Let us now define Note that the sum is over the set M r of all rooted planar maps.From the fact that Z q < ∞ it easily follows that Z (r) q < ∞.We then define the Boltzmann distribution B r q on the set M r by 2.2.The Brownian snake and the conditioned Brownian snake.Let x ∈ R. The Brownian snake with initial point x is a pair (b, r x ), where b = (b(s), 0 ≤ s ≤ 1) is a normalized Brownian excursion and r x = (r x (s), 0 ≤ s ≤ 1) is a real-valued process such that, conditionally given b, r x is Gaussian with mean and covariance given by for every 0 ≤ s ≤ s ′ ≤ 1.We know from [3] that r x admits a continuous modification.From now on we consider only this modification.In the terminology of [3] r x is the terminal point process of the one-dimensional Brownian snake driven by the normalized Brownian excursion b and with initial point x.
Write P for the probability measure under which the collection (b, r x ) x∈R is defined.As mentioned in [12], for every x > 0, we have We may then define for every x > 0 a pair (b x , r x ) which is distributed as the pair (b, r x ) under the conditioning that inf s∈[0,1] r x (s) ≥ 0.
We equip C([0, 1], R) 2 with the norm (f, g) = f u ∨ g u where f u stands for the supremum norm of f .The following theorem is a consequence of Theorem 1.1 in [7].
Theorem 2.1.There exists a pair (b 0 , r 0 ) such that (b x , r x ) converges in distribution as The pair (b 0 , r 0 ) is the so-called conditioned Brownian snake with initial point 0.
Theorem 1.2 in [7] provides a useful construction of the conditioned object (b 0 , r 0 ) from the unconditioned one (b, r 0 ).In order to present this construction, first recall that there is a.s. a unique s * in (0, 1) such that (see Lemma 16 in [9] or Proposition 2.5 in [7]).For every s ∈ [0, ∞), write {s} for the fractional part of s.According to Theorem 1.2 in [7], the conditioned snake (b 0 , r 0 ) may be constructed explicitly as follows : for every s 2.3.Statement of the main result.We first need to introduce some notation.Let m ∈ M r .We denote by o its root vertex.The radius R m is the maximal distance between o and another vertex of m that is The profile of m is the measure λ m on {0, 1, 2, . ..} defined by Note that R m is the supremum of the support of λ m .It is also convenient to introduce the rescaled profile.If m has n vertices, this is the probability measure on R + defined by Recall from section 2.2 that (b, r 0 ) denotes the Brownian snake with initial point 0.
Theorem 2.2.Let q be a regular critical weight sequence.There exists a scaling constant C q such that the following results hold.
(ii) The law of the random probability measure λ to the law of the random probability measure I defined by dt.
(iii) The law of the rescaled distance n −1/4 d(o, a) where a is a vertex chosen uniformly at random among all vertices of m, under B r q (• | #V m = n) converges as n → ∞ to the law of the random variable Theorem 2.2 is an analogue to Theorem 2.5 in [12] in the setting of non-bipartite maps.Beware that in Theorem 2.2 maps are conditioned on their number of vertices whereas in [12] they are conditioned on their number of faces.However the results stated in Theorem 2.5 in [12] remain valid by conditioning on the number of vertices (with different scaling constants).On the other hand, our arguments to prove Theorem 2.2 do not lead to the statement of these results by conditioning maps on their number of faces.A notable exception is the case of k-angulations (q = qδ k for some k ≥ 3 and appropriate q > 0), where an application of Euler's formula shows that #F m = (k/2 − 1)#V m + 2, so that the two conditionings are essentially equivalent and result in a change in the scale factor C q .
Recall that the results of Theorem 2.5 in [12] for the special case of quadrangulations were obtained by Chassaing & Schaeffer [2] (see also Theorem 8.2 in [4]).
Last observe that Theorem 2.2 is obviously related to Theorem 1 in [11].Note however that [11] deals with rooted pointed maps instead of rooted maps as we do and studies distances from the distinguished point of the map rather than from the root vertex.
2.4.Multitype spatial trees.We start with some formalism for discrete trees.Set where by convention N = {1, 2, 3, . ..} and N 0 = {∅}.An element of U is a sequence u = u 1 . . .u n , and we set |u| = n so that |u| represents the generation of u.In particular |∅| = 0.If u = u 1 . . .u n and v = v 1 . . .v m belong to U, we write uv = u 1 . . .u n v 1 . . .v m for the concatenation of u and v.In particular ∅u = u∅ = u.If v is of the form v = uj for u ∈ U and j ∈ N, we say that v is a child of u, or that u is the father of v, and we write u = v.More generally if v is of the form v = uw for u, w ∈ U, we say that v is a descendant of u, or that u is an ancestor of v.The set U comes with the natural lexicographical order such that u v if either u is an ancestor of v, or if u = wa and v = wb with a ∈ U * and b ∈ U * satisfying a 1 < b 1 , where we have set A plane tree t is a finite subset of U such that • for every u ∈ t there exists a number c u (t) ≥ 0 such that uj ∈ t ⇔ 1 ≤ j ≤ c u (t).
Let t be a plane tree and let ξ = #t − 1.The search-depth sequence of t is the sequence u 0 , u 1 , . . ., u 2ξ of vertices of t wich is obtained by induction as follows.First u 0 = ∅, and then for every i ∈ {0, 1, . . ., 2ξ − 1}, u i+1 is either the first child of u i that has not yet appeared in the sequence u 0 , u 1 , . . ., u i , or the father of u i if all children of u i already appear in the sequence u 0 , u 1 , . . ., u i .It is easy to verify that u 2ξ = ∅ and that all vertices of t appear in the sequence u 0 , u 1 , . . ., u 2ξ (of course some of them appear more that once).We can now define the contour function of t.For every k ∈ {0, 1, . . ., 2ξ}, we let C(k) = |u k | denote the generation of the vertex u k .We extend the definition of C to the line interval [0, 2ξ] by interpolating linearly between successive integers.Clearly t is uniquely determined by its contour function C.
Let K ∈ N and [K] = {1, 2, . . ., K}.A K-type tree is a pair (t, e) where t is a plane tree and e : t → [K] assigns a type to each vertex.If (t, e) is a K-type tree and if i ∈ [K] we set t i = {u ∈ t : e(u) = i}.We denote by T (K) the set of all K-type trees and we set and we set |w| = n.Consider the natural projection p : W → Z K + where p(w) = (p 1 (w), . . ., p K (w)) and Let u ∈ U and let (t, e) ∈ T (K) such that u ∈ t.We then define w u (t) ∈ W K by w u (t) = (e(u1), . . ., e(uc u (t))), and we set z u (t) = p(w u (t)).
A K-type spatial tree is a triple (t, e, ℓ) where (t, e) ∈ T (K) and ℓ : t → R. If v is a vertex of t, we say that ℓ v is the label of v.We denote by T (K) the set of all K-type spatial trees and we set If (t, e, ℓ) ∈ T (K) we define the spatial contour function of (t, e, ℓ) as follows.Recall that u 0 , u 1 , . . ., u 2ξ denotes the search-depth sequence of t.First if k ∈ {0, . . ., 2ξ}, we put V (k) = ℓ u k .We then complete the definition of V by interpolating linearly between successive integers.

Multitype spatial Galton
) be a family of probability measures on the set W K .We associate with ζ the family µ = (µ (i) , i ∈ [K]) of probability measures on the set Z K + in such a way that each µ (i) is the image measure of ζ (i) under the mapping p.We make the basic assumption that max and we say that ζ (or µ) is non-degenerate.If for every i ∈ [K], w ∈ W K and z = p(w) we have then we say that ζ is the uniform ordering of µ.
For every i, j be the mean number of type-j children of a type-i individual, and let The matrix M µ is said to be irreducible if for every i, j ∈ [K] there exists n ∈ N such that m (n) ij > 0 where we have written m Assume that ζ is non-degenerate, irreducible and (sub-)critical.We denote by ζ the law of a K-type Galton-Watson tree with offspring distribution ζ and with ancestor of type i, meaning that for every (t, e) ∈ T The fact that this formula defines a probability measure on T (K) i is justified in [10].
Let us now recall from [10] how one can couple K-type trees with a spatial displacement in order to turn them into random elements of T (K) .To this end, consider a family ν = (ν i,w , i ∈ [K], w ∈ W K ) where ν i,w is a probability measure on R |w| .If (t, e) ∈ T (K) and x ∈ R, we denote by R ν,x ((t, e), dℓ) the probability measure on R t which is characterized as follows.For every i ∈ [K] and u ∈ t such that e(u) = i, consider Y u = (Y u1 , . . ., Y u|w| ) (where we have written w u (t) = w) a random variable distributed according to ν i,w , in such a way that (Y u , u ∈ t) is a collection of independant random variables.We set L ∅ = x and for every v ∈ t \ {∅}, where ]]∅, v]] is the set of all ancestors of v distinct from the root ∅.The probability measure R ν,x ((t, e), dℓ) is then defined as the law of (L v , v ∈ t).We finally define for every x ∈ R a probability measure P 2.6.The Bouttier-Di Francesco-Guitter bijection.We start with a definition.We consider the set T M ⊂ T (4) 1 of 4-type trees in which, for every (t, e) ∈ T M and u ∈ t, 1. if e(u) = 1 then z u (t) = (0, 0, k, 0) for some k ≥ 0, 2. if e(u) = 2 then z u (t) = (0, 0, 0, 1), 1 be the set of 4-type spatial trees (t, e, ℓ) such that (t, e) ∈ T M and in which, for every (t, e, ℓ) ∈ T M and u ∈ t, We will be interested in the set Notice that condition 6. implies that if (t, e, ℓ) ∈ T M then ℓ v ≥ 0 for all v ∈ t.
We will now describe the Bouttier-Di Francesco-Guitter bijection from the set T M onto M r .This bijection can be found in [1] in the more general setting of Eulerian maps.
Suppose that the tree t is drawn in the plane and add an extra vertex ∂, not on t.We associate with (t, e, ℓ) a planar map whose set of vertices is and whose edges are obtained by the following device : for every k ∈ {0, 1, . . ., ξ − 1}, draw an edge between v k and the first vertex in the sequence v k+1 , . . ., v ξ with type 1 and label Notice that condition 6. in the definition of the set for every k ∈ {0, 1, . . ., ξ − 1}, and recall that min{ℓ v j : j ∈ {k + 1, . . ., ξ} and e(v j ) = 1} = 1.The preceding properties ensure that whenever e(v k ) = 1 and . The construction can be made in such a way that edges do not intersect.Notice that condition 2. in the definition of the set T M entails that a type-2 vertex is connected by the preceding construction to exactly two type-1 vertices with the same label, so that we can erase all type-2 vertices.The resulting planar graph is a planar map.We view this map as a rooted planar map by declaring that the distinguished edge is the one corresponding to k = 0, pointing from δ, in the preceding construction.
It follows from [1] that the preceding construction yields a bijection Ψ r between T M and M r .Furthermore it is not difficult to see that Ψ r satisfies the following two properties : let (t, e, ℓ) ∈ T M and let m = Ψ r ((t, e, ℓ)), (i) the set F m is in one-to-one correspondence with the set t 3 ∪ t 4 , more precisely, with every v ∈ t 3 (resp.v ∈ t 4 ) such that z u (t) = (k, k ′ , 0, 0) is associated a unique face of m whose degree is equal to 2k + k ′ + 2 (resp.2k + k ′ + 1), (ii) for every l ≥ 1, the set {a ∈ V m : d(∂, a) = l} is in one-to-one correspondence with the set {v ∈ t 1 : ℓ v = l}.
We set µ q = µ (1) q , µ and M µ q = (m ij ) 1≤i,j≤4 .The matrix M µ q is given by We see that M µ q is irreducible and has a spectral radius ̺ = 1.Thus µ q is critical.Let us denote by a = (a 1 , a 2 , a 3 , a 4 ) the right eigenvector of M µ q with eigenvalue 1 chosen so that a 1 + a 2 + a 3 + a 4 = 1.
The following proposition can be proved from Proposition 3 of [11] in the same way as Corollary 2.3 of [12].

A conditional limit theorem for multitype spatial trees
Let q be a regular critical weight sequence.Recall from section 2.7 the definition of the offspring distribution µ q associated with q and the definition of the spatial displacement distributions ν.To simplify notation we set µ = µ q .
In view of applying a result of [10], we have to take into account the fact that the spatial displacements ν are not centered distributions, and to this end we will need a shuffled version of the spatial displacement distributions ν.Let i ∈ [K] and w ∈ W. Set n = |w|.We set ← − w = (w n , . . ., w 1 ) and we denote by ← − ν i,w the image of the measure ν i,w under the mapping S n : (x 1 , . . ., x n ) → (x n , . . ., x 1 ).Last we set We write If (t, e, ℓ) is a multitype spatial tree, we denote by C its contour function and by V its spatial contour function.Recall that C([0, 1], R) 2 is equipped with the norm (f, g) = f u ∨ g u .The following result is a special case of Theorem 4 in [10].
Theorem 3.1.Let q be a regular critical weight sequence.There exists two scaling constants A q > 0 and B q > 0 such that for i ∈ {1, 2}, the law under converges as n → ∞ to the law of (b, r 0 ).The convergence holds in the sense of weak convergence of probability measures on C([0, 1], R) 2 .
Note that Theorem 4 in [10] deals with the so-called height process instead of the contour process.However, we can deduce Theorem 3.1 from [10] by classical arguments (see e.g.[5]).Moreover, the careful reader will notice that the spatial displacements ← → ν depicted above are not all centered, and thus may compromise the application of [10,Theorem 4].However, it is explained in [11,Sect. 3.3] how a simple modification of these laws can turn them into centered distributions, by appropriate translations.More precisely, one can couple the spatial trees associated with ← → ν and its centered version so that the labels of vertices differ by at most 1/2 in absolute value, which of course does not change the limiting behavior of the label function rescaled by n −1/4 .
In this section, we will prove a conditional version of Theorem 3.1.Before stating this result, we establish a corollary of Theorem 3.1.To this end we set µ, ← → ν ,0 (dt de dℓ | c ∅ (t) = 1).Notice that this conditioning makes sense since µ (1) ({(0, 0, 1, 0)}) > 0. We may also define for every n ≥ 1, The following corollary can be proved from Theorem 3.1 in the same way as Corollary 2.2 in [12].Corollary 3.2.Let q be a regular critical weight sequence.The law under converges as n → ∞ to the law of (b, r 0 ).The convergence holds in the sense of weak convergence of probability measures on C([0, 1], R) 2 .
Recall from section 2.2 that (b 0 , r 0 ) denotes the conditioned Brownian snake with initial point 0.
Theorem 3.3.Let q be a regular critical weight sequence.For every x ≥ 0, the law under P (1),n converges as n → ∞ to the law of (b 0 , r 0 ).The convergence holds in the sense of weak convergence of probability measures on C([0, 1], R) 2 .
In the same way as in the proof of Theorem 3.3 in [12], we will follow the lines of the proof of Theorem 2.2 in [4] to prove Theorem 3.3.

3.1.
Rerooting spatial trees.If (t, e) ∈ T M , we write ∂t = {u ∈ t : c u (t) = 0} for the set of all leaves of t, and we write ∂ 1 t = ∂t ∩ t 1 for the set of leaves of t which are of type 1.Let w 0 ∈ t.Recall that U * = U \ {∅}.We set and we write e (w 0 ) for the restriction of the funtion e to the truncated tree t (w 0 ) .Let v 0 = u 1 . . .u 2p ∈ U * and (t, e) ∈ T M such that v 0 ∈ t 1 .We define k = k(v 0 , t) and l = l(v 0 , t) in the following way.Write ξ = #t − 1 and u 0 , u 1 , . . ., u 2ξ for the search-depth sequence of t.Then we set k = min{i ∈ {0, 1, . . ., 2ξ} : which means that k is the time of the first visit of v 0 in the evolution of the contour of t and that l is the time of the last visit of v 0 .Note that l ≥ k and that l = k if and only if v 0 ∈ ∂t.For every s ∈ [0, 2ξ − (l − k)], we set where C is the contour function of t and Then there exists a unique plane tree t (v 0 ) whose contour function is C (v 0 ) .Informally, t (v 0 ) is obtained from t by removing all vertices that are descendants of v 0 , by re-rooting the resulting tree at v 0 , and finally by reversing the planar orientation.Furthermore we see that v 0 = 1u 2p . . .u 2 belongs to t (v 0 ) .In fact, v 0 is the vertex of t (v 0 ) corresponding to the root of the initial tree.At last notice that c ∅ ( t And ζ (e(u)) (w u (T )).
By the above discussion around (2), the terms corresponding to u, u in these two products are all equal, except for those corresponding to vertices then the spatial displacements between u and its offspring is left unchanged by the re-rooting, meaning that ], set j = j(u, v 0 ), and define the mapping φ n,j : (x 1 , . . ., x n ) → (x j−1 − x j , . . ., x 1 − x j , −x j , x n − x j , . . ., x j+1 − x j ) .
Then observe that the spatial displacements are affected in the following way: If (t, e, ℓ) ∈ T M and w 0 ∈ t, we also consider the multitype spatial tree (t (w 0 ) , e (w 0 ) , ℓ (w 0 ) ) where ℓ (w 0 ) is the restriction of ℓ to the tree t (w 0 ) .Recall the definition of the probability measure Q µ, ← → ν .Lemma 3.5.Let v 0 ∈ U * be of the form v 0 = 1u 2 . . .u 2p for some p ∈ N. Assume that Then the law of the re-rooted multitype spatial tree ( t (v 0 ) , e (v 0 ) , ℓ (v 0 ) ) under the measure coincides with the law of the multitype spatial tree (t This lemma is a simple consequence of Lemma 3.4 and our observations around (4) on the spacial displacements ← → ν , combined with the discussion of how the set of children of various vertices are affected by re-rooting, see (1) and (2).Lemma 3.6.Let w ∈ W such that p 3 (w) = p 4 (w) = 0. Set n = |w| and let j ∈ {1, . . ., n}.
We consider a uniform vector (X l + ½ {w l−1 =1} , 1 ≤ l ≤ n + 1) on the set A k,k ′ and we set X (j) = (X j+1 , . . ., X n+1 , X 1 , . . ., X j ).Since w 0 = w j = 1, the vector and the measure ν 3,w j,1 is the law of the vector Furthermore we notice that X 1 + X 2 + . . .+ X n+1 = 0.This implies that which means that the measure ν 3,w j,1 is the image of ν 3,w under the mapping φ n,j .Since φ n,j • S n = S n • φ n,n−j+1 , we obtain together with what precedes that the measure w under the mapping φ n,j .Thus ← → ν 3,w j,1 is the image of ← → ν 3,w under the mapping φ n,j .Hence, it is the image of the same measure under φ n,j , being invariant under the action of S n .Thus we get the first part of the lemma.The other assertions can be proved in the same way.
We also denote by v m the first element of ∆ 1 in the lexicographical order.
The following two Lemmas can be proved from Lemma 3.5 in the same way as Lemma 3.3 and Lemma 3.4 in [4].Lemma 3.7.For any nonnegative measurable functional
We will now state a lemma which plays a crucial role in the proof of the main result of this section.To this end, recall the definition of v m and the definition of the probability measure Q n µ, ← → ν .Lemma 3.11.There exists a constant c > 0 such that for all n sufficiently large, Proof : We first treat the case where q 2k+1 = 0 for every k ≥ 2 which implies that q 3 > 0.

3 2
Figure 1.The branch leading from ∅ to v 0 , and the corresponding branch in the tree t (v 0 ) : the branch is put upside-down and the vertices of [[∅, v 0 ]] 3 2